I have seen pictures of clusters of galaxies, usually used in regards to theories of dark matter and galaxy formations. One of the most famous ones has the perceived shape of a stick-figure. If I am not mistaken some of these clusters seem to be bigger than our observable Universe horizon of $14$ billion light-years, which is as far as we can see because of the age of the Universe.

My assumption is that these are just coordinates plotted into a computer simulation. However, are these clusters the size of our observable Universe horizon or are these more theoretical abstractions?

2 Answers
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The scale of galaxy clusters are on the order of ~1Mpc (~3.14 Million light years in size), and are therefore much smaller than the cosmological horizon.

The cosmological principle is extraordinarily important for cosmologists, and makes the assumption about the universe's global properties (which turn out to be really good assumptions, so far):

Homogeneity on the scale of about 100 Mpc. This means that one would see very little change in density within a bubble of radiua 100 Mpc as you move said bubble around the universe.

Isotropy means that there is no preferred directions to look. No matter where you look, you should see roughly the same picture of the universe.

As for how these clusters are generated (addressing the title of your question), large N-body simulations are run over the age of the universe from different sets of initial conditions (this is where different models and assumptions come into it). These are dark matter only simulations and involve only the force of gravity. Some people are including electromagnetic interactions in their code, but it's far from being the norm, and are really only important on small scales, i.e. - inner regions of galaxies and clusters. These simulations can contain upwards of ~10 billion dark matter particles.

No, the structures are not of the size of the observable Universe. In fact modern cosmolgical models rely on the Universe being homogeneous over scales above $100 \textrm{Mpc}$. This is an observational fact, which serves as a base for the so-called cosmological principle. See also a wiki article on large-scale structure, where the scale is mentioned in the section "End of Greatness".

The scale corresponds to a few hundred million light years. The largest structures, therefore, are clearly way smaller than the size of the observable Universe, which is of order of ten billion light years.

Note, that because it takes time for the light to reach us, the current size of the Universe we observe is larger than its apparent size due to cosmological expansion.